Tchouameni has joined up with the France national team squad ahead of their UEFA Nations League games in the coming weeks despite the ongoing transfer speculation on his future.
The 22-year-old has been heavily linked with a move away from France ahead of the 2022/23 season amid growing interest from Los Blancos.
According to reports from French outlet RMC Sport, via Diario AS, negotiations are already well into their final stages, with a deal potentially wrapped up before Tchouameni returns for pre-season training in July.
The report adds Tchouameni rejected a move to Ligue 1 rivals Paris Saint-Germain despite Les Bleus teammate Kylian Mbappe personally reaching out to him over a move to the French capital.
However, the challenge of playing outside of France has been a key motivator for Tchouameni, with Real Madrid edging out Liverpool in the race for the former Bordeaux schemer.